The Height fluctuations of an off-critical dimer model on the square grid

The dimer model on a planar bipartite graph can be viewed as a random surface measure. We study these fluctuations for a dimer model on the square grid with two different classes of weights and provide a condition for their equivalence. In the thermodynamic limit and scaling window, these height fluctuations are shown to be non-Gaussian. They are also rotationally invariant for a certain choice of weights. Finally, we show that the height difference between any two points is Gaussian for a specific choice of weights.